{"id":"1b26faae-f354-4c8e-903a-d3b6b8f716dd","arxiv_id":"2607.11246","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"Weak regular inference functions and Stein instruments induce Godambe Riemannian metrics on parametric models far outside the Fisher–Rao class, including undominated Cantor location families and α-stable lattice SPDEs.","lead":"This paper shows that many statistical models without densities or Fisher information can still be treated as Riemannian manifolds by probing laws with estimating-function instruments. That lets geometric tools reach undominated, support-dependent, and heavy-tailed models that classical information geometry excludes.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the reader's already-flagged Assumption 2.3; the central claim holds under its stated premises.","rationale":"The central claim of Theorem 1.1 is a clean, correctly proved extension once the distributional framework and Assumption 2.3 are accepted as external premises. The five examples supply concrete instruments with closed-form S and V, so the hypotheses are not vacuous. The reader's CONDITIONAL verdict already captures the only material caveats (assumed weak differentiability, companion dependence, unbacked numerics). No stronger load-bearing concern—hidden contradiction, mis-stated Loewner order, or failure of the metric axioms—surfaces on a second pass. The recommended concrete check is a verification of the most elaborate closed-form example rather than a challenge to the logic. Verdict therefore remains CONDITIONAL with no adjustment.","tokens_in":34401,"tokens_out":617,"duration_ms":7200,"concrete_test":"Independently recompute the two-probe Godambe information (18) for the lattice SPDE at a fixed (N,θ,α,k,l,c,d) by numerical quadrature of the joint characteristic-function integrals (16)–(17) and compare against the Monte-Carlo estimate of SᵀV⁻¹S obtained from Euler–Maruyama paths of (6); if the relative discrepancy exceeds a few percent after accounting for Monte-Carlo error, the closed-form claims of Section 5 (and thus the spectral-gap rate) would be unreliable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption (Assumption 2.3: weak differentiability of θ ↦ ⟨T_θ,g⟩ with derivatives passing inside the pairing) is correctly identified as the analytic hinge for the weak Bartlett identity (1), for S(θ)=⟨∂_θ T_θ,ψ⟩, and thus for smoothness of G in Theorem 1.1 / Proposition 3.1. Once that premise and the existence of an instrument with full-rank S and positive-definite V are granted, the remainder of the argument is standard linear algebra (positive-definiteness of SᵀV⁻¹S) plus the classical Loewner comparison of Proposition 3.2. The paper does not claim a general existence theorem for instruments or for Assumption 2.3; it exhibits both explicitly in five examples (uniform, shifted exponential, Cantor location, stratified mixture, lattice α-stable SPDE) and defers genericity to the companion [25]. No internal inconsistency appears: non-canonicity of G is acknowledged (Section 7, Remark 7.1), the Fisher recovery and Loewner order are correctly stated, and the spectral-gap rate in Proposition 5.1 follows by direct Taylor expansion of the closed-form characteristic-function expressions. The dependence on same-author companions is a presentational limitation, not a correctness flaw in the derivation as written.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper argues that parametric models represented by tempered distributions T_θ ∈ S'(R^k) can be equipped with Riemannian metrics far beyond the classical Fisher–Rao setting. An external instrument (weak regular inference function or weak Stein representation) with full-rank sensitivity S and positive-definite variability V induces the Godambe information G = S^T V^{-1} S as a smooth metric on Θ (Theorem 1.1 / Proposition 3.1). Fisher–Rao is recovered when the score is admissible, and G ⪯ I in Loewner order whenever Fisher exists (Proposition 3.2). Five examples outside Fisher–Rao are treated in closed form: uniform scale, shifted exponential, undominated Cantor location, stratified mixture with biased score, and a lattice α-stable heat equation whose Godambe metric stabilises at the spectral-gap rate. Quadratic Stein discrepancies recover the same local geometry; non-canonicity of the instrument and block-diagonality as geometric nonformation are discussed.","tokens_in":34731,"tokens_out":1114,"duration_ms":10528,"significance":"If the construction is accepted, the class of models that can be treated by differential-geometric methods expands substantially: undominated families, parameter-dependent support, moment-free laws, and dominated models with biased scores all become Riemannian. The closed-form metrics (uniform G=3/θ², Cantor G≡8, SPDE spectral-gap rate αθλ_k) and the explicit Loewner comparison are concrete, usable contributions. The paper is honest about non-canonicity (Section 7, Remark 7.1) and about the instrument-relative character of the geometry. Dependence on the author’s 2026 companion series for the full apparatus of weak inference functions and genericity is a presentational limitation rather than a correctness flaw in the derivations as written.","major_comments":[{"comment":"Assumption 2.3 (weak differentiability of θ ↦ ⟨T_θ, g⟩ with derivatives passing inside the pairing) is the analytic hinge for the weak Bartlett identity (1), for S(θ)=⟨∂_θ T_θ, ψ⟩, and thus for smoothness of G in Theorem 1.1 / Proposition 3.1. It is stated as an assumption and verified only by direct computation in the five examples; no general criterion is given for when a tempered-distribution model map satisfies it. For the central claim to be usable beyond the exhibited cases, the paper should either supply verifiable sufficient conditions (e.g., dominated differentiability of characteristic functions, or continuity of the map in the weak-* topology of S') or state more explicitly that the theorem is conditional on this hypothesis and that existence is left to the companion [25].","section":null},{"comment":"Proposition 1.2 and the examples establish existence of instruments case-by-case, but the paper repeatedly speaks of “the class of Godambe–Riemannian models” as if it were characterised. Without a general existence theorem (or a clear statement that none is claimed), the scope of the extension remains an open-ended collection of examples. A short subsection clarifying what is proved versus what is exhibited would prevent over-reading of Theorem 1.1.","section":null}],"minor_comments":[{"comment":"Heavy self-citation to the 2026 companion series ([21]–[25]) makes the paper hard to read in isolation. A short self-contained appendix restating Definition 2.2 and the admissible classes of Lemma 2.1 would help.","section":null},{"comment":"Notation: α and β are used both for the stability index / eigenmode scales (Section 5) and for interest/nuisance parameters (Section 8). A local warning is present but easy to miss; consider distinct symbols.","section":null},{"comment":"Remark 4.1 (Student-t / Cauchy) and Remark 4.9 (Cauchy mixture) are useful but sit outside the main example sections; a pointer in the introduction would improve navigation.","section":null},{"comment":"Typographical: “F amily of metrics” in the Contents (Section 7 heading) has a stray space; “P´ olya” and similar accented names appear inconsistently.","section":null},{"comment":"Section 5.6 claims numerical verification of the closed-form CF and of Proposition 5.1 against Monte Carlo, but no figure or table is supplied. A brief plot of |G_k(t)−G_k(∞)| versus the predicted rate would strengthen the claim.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is part of a tightly interlinked 2026 arXiv series by the same author. The technical core (Propositions 3.1–3.2, closed-form examples) stands on its own, but the journal may wish to confirm that the companion papers are either already public or will be made available so that referees and readers can check the deferred genericity and inference-function apparatus. Scope fit for a pure math.ST journal is good; the SPDE example is a genuine plus."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is that Labouriau gets Riemannian metrics on models where Fisher–Rao never starts—undominated Cantor location, parameter-dependent support, biased-score mixtures, α-stable lattice heat—by treating the law as a tempered distribution and reading geometry off an external instrument (weak inference function or Stein residual). Theorem 1.1 and Props 3.1–3.2 are the classical Godambe sandwich plus Loewner projection of the score; once full-rank S and PD V are granted, the rest is linear algebra. That is not a defect: the paper is honest that G is instrument-relative and that Chentsov uniqueness is traded for existence.\n\nWhat is new is the packaging and the worked examples. The Cantor family (Prop 4.2: no dominating measure at all) with closed-form flat metrics from moments and transform residuals is the cleanest illustration. The stratified Gaussian mixture recovers a metric where the score is biased. The lattice SPDE gives an explicit spectral-gap rate for stabilisation of G (Prop 5.1) and a collapse under unstable dynamics—nice bridge between dynamical and geometric stability without needing variance. Stein–Godambe equivalence for quadratic discrepancies and the automatic block-diagonality for odd/even probes in symmetric location-scale models (including Cantor) are clean structural payoffs.\n\nSoft spots are real but proportionate. Assumption 2.3 (weak differentiability of the model map) is the analytic hinge and is assumed, not proved for general T_θ; instrument existence is exhibited, not generally proved (genericity deferred to companion [25]). Heavy self-citation to the 2026 companion series is a presentational tax, not a circularity in the sandwich itself. SPDE numerics are claimed but not artifact-backed. Free parameters (probe frequencies) are acknowledged; non-canonicity is discussed rather than papered over.\n\nThis is for people who already care about estimating-function geometry, non-regular models, or Stein discrepancies and want usable metrics and distances outside the Fisher–Rao class. The math is solid under its premises; citations to Godambe, Amari–Kawanabe, Chentsov, and the kernel-Stein literature are appropriate. I would send it to peer review. Worth engaging if you work in that corner of math.ST.","headline":"Solid, usable extension of information geometry via Godambe instruments on tempered distributions; examples carry the weight and the math is clean under stated premises.","tokens_in":35366,"tokens_out":576,"would_cite":true,"duration_ms":7289,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62B11","62F10","53C20","60G52"],"pacs":[],"model":"grok-4.5","headline":"Parametric models without densities, scores, or Fisher information can still carry Riemannian metrics once laws are tempered distributions and instruments extract Godambe information.","keywords":["information geometry","Godambe information","tempered distributions","weak inference functions","Stein discrepancies","Fisher–Rao manifold","nonformation","alpha-stable noise"],"falsifier":"Exhibit a concrete parametric family of tempered distributions together with an instrument whose sensitivity and variability are smooth, full-rank and positive-definite, yet the resulting G fails to be a Riemannian metric, or show that the claimed Loewner domination G ⪯ I fails when Fisher information exists and the Bartlett interchange holds.","tokens_in":35229,"feed_emoji":"📐","tokens_out":1034,"duration_ms":14282,"temperature":0.7,"pith_summary":"Classical information geometry needs densities, scores, and finite Fisher information, so many models—singular continuous laws, parameter-dependent supports, transform-only specifications, and even some dominated mixtures with biased scores—are left outside the Riemannian picture. This paper replaces densities by tempered distributions and extracts information with external instruments (weak regular inference functions or weak Stein representations). Any instrument with full-rank sensitivity and positive-definite variability induces a smooth Godambe metric on the parameter space; the Fisher–Rao metric is recovered exactly when the score itself is admissible, and every Godambe metric sits below the Fisher metric in the Loewner order whenever the latter exists. Five concrete examples, including a Cantor location family that admits no dominating measure and a lattice heat equation driven by alpha-stable noise, show that the construction works where Fisher–Rao cannot even begin. Because no instrument is canonical, a model carries a family of metrics whose members serve different inferential, diagnostic, geometric and computational roles, while weak inferential separation appears as block-diagonality of the metric.","feed_headline":"Models without densities still carry Riemannian metrics","feed_subtitle":"Instruments on tempered distributions induce Godambe geometry where Fisher–Rao cannot start","key_machinery":"The Godambe information G = Sᵀ V⁻¹ S built from the weak sensitivity and variability of an instrument acting on tempered distributions; it is the metric tensor that turns the parameter space into a Godambe–Riemannian manifold and recovers Fisher information when the instrument is the score.","core_discovery":"Once a parametric family is represented by tempered distributions T_θ, any instrument (weak regular inference function or weak Stein representation) whose sensitivity S is full-rank and whose variability V is positive definite induces the Godambe information G(θ) = S(θ)ᵀ V(θ)⁻¹ S(θ) as a smooth Riemannian metric on the parameter space. The Fisher–Rao manifold is the special case in which the score is itself an admissible instrument; whenever Fisher information exists, every Godambe metric is dominated by it in the Loewner order.","pith_inferences":["The same instrument construction could be applied to other singular or heavy-tailed SPDE models to obtain closed-form geometric stability rates without second-moment assumptions.","Hierarchy of RKHS Stein geometries suggests a practical route to adaptive metric selection: start with low-frequency or moment instruments and enrich only when diagnostics show near-degeneracy.","Automatic block-diagonality for odd/even probes in every symmetric location-scale family (including Cauchy and Cantor) supplies an immediate, likelihood-free method for exact location–scale separation in robust estimation."],"forward_implications":["Undominated families such as the Cantor location model, models with parameter-dependent support, and mixtures whose score is biased all become Riemannian manifolds once a suitable instrument is chosen.","Quadratic Stein discrepancies built from finite collections of identities induce exactly the same local geometry as the corresponding Godambe metrics.","Weak inferential separation (nonformation) is equivalent to block-diagonality of the Godambe metric with respect to the interest–nuisance splitting.","In dynamical models such as the lattice stochastic heat equation driven by alpha-stable noise, the Godambe geometry stabilises at a rate controlled by the spectral gap of the discrete Laplacian.","Because there is no canonical instrument, a single model carries a family of metrics that can be selected according to inferential, diagnostic, geometric or computational purpose."],"fun_headline_variants":["Instruments on tempered distributions induce Godambe Riemannian metrics","Godambe geometry extends Fisher-Rao via weak inference instruments","Models without densities gain metrics from distributional instruments","Weak Stein instruments yield Riemannian structure on parameter space","Godambe metrics from full-rank instruments on tempered distributions"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The map from parameters to the distributional representation must be weakly differentiable: for every admissible probe the expectation must be continuously differentiable and derivatives must pass inside the pairing.","fun_headline_variants_meta":{"raw":{"variants":["Instruments on tempered distributions induce Godambe Riemannian metrics","Godambe geometry extends Fisher-Rao via weak inference instruments","Models without densities gain metrics from distributional instruments","Weak Stein instruments yield Riemannian structure on parameter space","Godambe metrics from full-rank instruments on tempered distributions"]},"model":"grok-4.5","effort":"low","cost_usd":0.004054,"raw_usage":{"total_tokens":1357,"prompt_tokens":926,"num_sources_used":0,"completion_tokens":62,"cost_in_usd_ticks":40540000,"prompt_tokens_details":{"text_tokens":926,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":369,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":926,"tokens_out":62,"duration_ms":4778,"temperature":1.0,"reasoning_tokens":369,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T06:01:56.031341+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a concrete parametric family of tempered distributions together with an instrument whose sensitivity and variability are smooth, full-rank and positive-definite, yet the resulting G fails to be a Riemannian metric, or show that the claimed Loewner domination G ⪯ I fails when Fisher information exists and the Bartlett interchange holds.","supporting_citations":[],"review_version":1}